Task. Using the compass and straightedge, construct to a given circle the tangent lines through a given point outside the circle.

Solution. Let be the centre of the given circle and the given point. With as diameter, draw the circle (see midpoint). If and are the points where this circle intersects the given circle, then by Thales' theorem, the angles and are right angles. According to the definition of the tangent of circle, the lines and are required tangents.

The convex angle is called a tangent angle (or tangent-tangent angle) of the given circle and the convex angle the corresponding central angle. It is apparent that a tangent angle and the corresponding central angle are supplementary.

The tangent angle is the angle of view of the line segment from the point .

Note that if a circle is inscribed in a polygon, then the angles of the polygon are tangent angles of the circle and the centre of the circle is the common intersection point of the angle bisectors.